An automobile is modeled as a multi degree of freedom system as shown in Figure,the free body diagram and equations of motion of the Automobile suspension system is given.By using the Given Data Develop an analytical solution to predict the vibration responses of the multi-degree-of-freedom suspension system, considering the interaction between the vehicle's body, springs, shock absorbers, and road surface variations. Utilize classical analytical methods, matrix iteration method, and eigenvalue analysis to model the system's dynamic behavior. Find all the Natural Frequencies and Mode Shapes of this system.Start by converting the given equations into matrix form and continue from there.Incorporate varying road profiles and loading conditions that the vehicle encounters during operation. Investigate how changes in system parameters (such as spring rates, damping coefficients, or mass distributions) affect the system's vibration characteristics.

Free Body Diagram

Equations of motion:

Eqs. (1) to (4) can be rewritten as M3x3+2c2x3c2x1c2x2+2k2x3+(k22k21)k2x1k2x2=0 JG+(c212+c222)+(c21+c22)x3+c21x1c22x2+(k21+k22)x3+k21x1k22x2+(k212+k222)=0m1x1+(c1+c2)x1c2x3+c21+(k1+k2)x1k2x3+k21=F1m2x2+(c1+c2)x2c2x3c22+(k1+k2)x2k2x3k22=F2